2 TOPOLOGICAL CONCEPTS 5 (I) Au = EA , and hu is the identity mapping 1 EA ; (II) for each triple of indices (A, //, v) and each x e AA/4 n AAv , we have h^(x) e A^ and (12.2.4.1) M*) = MW*)) (patching condition). Let F be the M/W of the EA , which are therefore pairwise disjoint subsets of F (1.8). In F, consider the relation R(x, y) :" there exist A, \JL such that x e AA// , y e AMA , and y = M*)-" This is an equivalence relation. It is reflexive by virtue of condition (I) ; it is symmetric because h^ and h^ are inverses of each other (by applying (II) with v = A, and then (I)) ; finally, it is transitive, because if , y- ^x e ^ n MV , z = ^y then we have x = h^(y) and therefore x e AA/I n AAv by (II), and it follows now from (12.2.4.1 ) that z = hv^(x). We remark also that, by condition (I), the intersection of an EA and an equivalence class of R consists of at most one point. If E = F/R is the set of equivalence classes of R and if TT : F -> F/R = E is the canonical mapping, then each of the restrictions TTA = n | EA : EA -> E is injective. Moreover, the sets ?rA(EA) form a covering of E. Now consider the set O of subsets X of E with the following property: for each A e L, the set n^1(X n 7rA(EA)) is open in EA . It is clear that O satisfies axioms (Oj) and (On), and is therefore a topology on E. We shall show that in this topology the sets 7cA(EA) are open in E and that TCA is a homeomorphism of EA onto the subspace 7tA(EA) of E, for each A e L. In view of the definition of O and of the fact that TTA is a bijection of EA onto 7rA(EA), it is enough to establish the equivalence of the following two properties of a subset XA of EA : (a) XA is an open set in the topological space EA; (b) for each n e L, the set n~ 'I(TTA(XA) n n^E^)) is open in E^. Taking ju = A, we see that (b) implies (a). Conversely, if (a) is satisfied, then since we have n^1(n^(E^) n ^(E^)) = A^ by definition, the condition (b) signifies that, for each ju e L, the set /z^A(XA n AA)J is open in E^ . Now XA n AA/I is open in AA/i , and A^A is a homeomorphism of AA/i onto an open subset of EM. Hence (a) implies (b). The topological space E so defined is said to be obtained by patching together the EA along the AA// by means of the h^ . if EI denotes the topological space obtained by giving E the topology